Combining matrix transformations

If A acts first and B acts second, a point v becomes B(Av) = (BA)v. Matrix multiplication is associative but usually not commutative. Work out the product in the stated order before applying it to a column vector, or track a test point through both transformations to check the order.

For AQA 8365, rotations are multiples of 90° about the origin; the listed reflection lines are the axes and y = ±x; enlargements are centred on the origin. Translation cannot be represented by an ordinary 2 × 2 matrix acting on position vectors. A transformation matrix is determined by the images of (1,0) and (0,1), which become its first and second columns.

Worked example

Reflect in the x-axis, then rotate 90° anticlockwise. Find the combined matrix.

1. The reflection matrix is A = [[1,0],[0,−1]].
2. The rotation matrix is B = [[0,−1],[1,0]].
3. Multiply BA, taking each row of B against each column of A.

Answer: [[0,1],[1,0]], a reflection in y = x

Revise first: Matrix multiplication, Matrix transformations.